The problem of embedded eigenvalues for the Dirac equation in the Schwarzschild black hole metric

0.21 MB·

by  Davide Batic

The problem of embedded eigenvalues for the Dirac equation in the Schwarzschild
black hole metric
D. Batic
∗
Department of Mathematics,
The Petroleum Institute, Abu Dhabi, UAE
M. Nowakowski
†
Departamento de Fisica,
Universidad de los Andes,
Cra.1E No.18A-10, Bogota, Colombia
K. Morgan
‡
7 Department of Mathematics,
1 University of the West Indies, Kingston 6, Jamaica
0 (Dated: January 17, 2017)
2
WeusetheDiracequationinafixedblackholebackgroundanddifferentindependenttechniquesto
n
demonstratetheabsenceoffermionicboundstatesaroundaSchwarzschildblackhole. Inparticular,
a
weshowthatnoembeddedeigenvaluesexistwhichhavebeenclaimedforthecasewhentheenergyis
J
lessthantheparticle’smass. Weexplicitlyprovethattheclaimsregardingtheembeddedeigenvalues
4
canbetracedbacktoanoversimplifiedapproximationinthecalculation. Weconcludethatnobound
1
states exist regardless the valueof the mass.

]
c PACSnumbers: ValidPACSappear here
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-
r
g
[
I. INTRODUCTION

1
v
Inthe lastfifteenyearstheproblemofwhetherornotboundstatesexistfortheDiracequationinthepresenceofa
9
8 blackholeattractedalotofattentioninthescientificcommunity. Theexistingliteratureonthistopicischaracterized
8 by disagreeingresults. For instance, [1, 2] showedthat bound states existfor a Fermionimmersed in the geometryof
3 a Schwarzschildblack hole. Their argumentrelies on an approximationof the radialsystem emerging from the Dirac
0 equation after separationof variables and on the construction of approximatedsolutions at the event horizonand far
.
1 away from the black hole. Both of them end up with an approximated spectrum resembling that of a hydrogen-like
0 atom. On the other hand, [3] proved that such bound states cannot exist and only resonances are admitted. In their
7 approach the problem is tackled numerically after writing the Dirac equation in the Schwarzschild black hole (BH)
1 metric withrespecttoa gaugethatis well-suitedto numericalsolution. Furthermore,itasbeenshownin[4]thatthe
:
v Dirac equationdoes notadmit normalizable,time-periodic solutions ina non-extreme Reissner-Nordstro¨mblack hole
i geometrywheretheproofreliesontheHeisenbergUncertaintyPrincipleandtheparticularformofthe Diraccurrent.
X
A non-existence theorem regarding bound states for the Dirac equation in a Kerr-Newman geometry was obtained
r by [5]where the authorsintroducedcertainmatching conditions for the spinorfield acrossthe horizonswhichin turn
a
gave rise to a weak solution of the Dirac equation in the physical region of the maximal analytic extension of the
Kerr-Newman solution. More precisely, they exploited the conservation and positivity of the Dirac current to show
that because of the matching conditions, the only way in which a bound state solution of the Dirac equation can be
normalizableisthateachtermintheFourierexpansionofthespinorfieldintimeandtheangularvariablearoundthe
axis of symmetry, be identically zero. The case of the extreme Kerr metric has been treated by [6] where necessary
and sufficient conditions for the existence of bound states have been derived. However, since such conditions are
expressedinterms ofa complicatedsetofequationsandinequalities,the author didnotfurtherstudy the problemof
whether or not there exist values of the particle energy satisfying the aforementioned conditions. Instead of dealing
with the conditions derivedby [6], [7] provedthe non-existence of bound states for the Dirac equationin the extreme
Kerrgeometrybyapplyingthe so-calledIndexTheoremforDirac systemspresentedin[8]. Finally,the non-existence
Electronicaddress: [email protected]
∗
Electronicaddress: [email protected]
†
Electronicaddress: [email protected]
‡
2
of bound states for the Dirac equation in the presence of a Kerr-Newman-de Sitter black hole was showed in [9]
adopting an operator theory approach. In the present work we prove that no bound states occur for the Dirac
equationintheSchwarzschildBHmetric. Ourmethodisdifferentfromthoseappliedintheexistingliteratureonthis
topic and in a certain sense much simpler since it boils down to the analysis of a four-term recurrence relation. Our
paper is structured as follows. In Section I we give a short introduction motivating the importance of our findings.
In Section II we introduce the Dirac equation in the Schwarzschild BH metric and we review the different forms
(tetrad dependent) obtained for the radial system emerging after the Chadrasekharansatz has been applied. At this
point a comment is in order. The completeness of the Chandrasekhar ansatz has been proved for the first time in
[10] where an integral representation for the Dirac propagator in terms of the solutions of the radial and angular
ODEs arising by means of the Chandrasekhar method of separation of variables has been derived. [10] inferred the
completenessofthisansatzfromtheaforementionedintegralrepresentation. Analternativeproofofthecompleteness
of the Chandrasekhar ansatz can also be found in [11]. In Section III rather than approximating the radial system
as in [1, 2], where bound states were found due to the oversimplification of the problem, we decouple it and show
that eachradialspinor satisfies a generalizedHeun equation[12–14]. Then, we require that the radialspinors exhibit
exponential decay asymptotically away from the black hole. This will be always the case if ω2 < m2 where ω is the
e
energy of the particle and m the mass of the Fermion. The next step consists in establishing whether or not it is
e
possibletofindrepresentationsoftheradialwavefunctionsconsistingofanexponentialdecayingatspace-likeinfinity
multiplied by a polynomial function. The presence of a polynomial function will ensure integrability at the event
horizon. Similarlytothe study ofboundstatesforthe hydrogenatomwe employapowerseriesrepresentationwhose
coefficients turn out to satisfy a four-term recurrence relation instead of a two-term recurrence relation as it is the
case for the hydrogen atom. At this point we derive necessary and sufficient conditions ensuring that our recurrence
relation breaks down in the sense that the first N coefficients of the power series expansion do not vanish and the
subsequent coefficients are all zero. Finally, we show by contradiction that there exist no real values of the energy of
theFermionsatisfyingtheaforementionedconditionsandhence,noboundstatescanoccur. However,ifoneallowsfor
complexvaluesofthe energy,thenourconditionscanbesatisfiedsimultaneouslyandresonancesappearinagreement
with the results in [3]. InSection IV we give analternativeproofrelying onthe applicationofthe Index Theoremfor
Dirac systems (see [8]).
II. THE DIRAC EQUATION IN THE SCHWARZSCHILD BH METRIC
The Schwarzschildspace-time represents a static black hole immersed in an asymptotically Minkowski space-time.
In coordinates (t,r,ϑ,ϕ) with r>0, 0 ϑ π, 0 ϕ<2π the SchwarzschildBH metric is given by [15]
≤ ≤ ≤
1
ds2 = 1 2M dt2 1 2M − dr2 r2(dϑ2+sin2ϑdϕ2), (1)
− r − − r −
(cid:18) (cid:19) (cid:18) (cid:19)
where M is the mass of the black hole. A fermion of mass m and charge e in a curved space-time is described by
e
the Dirac equation (~=c=G=1) [16, 17]
m
∇AA′PA+iµ∗QA′ =0, ∇AA′QA+iµ∗PA′ =0, µ∗ = √2e, (2)
where ∇AA′ denotes covariant differentiation, and (PA,QA′) with A = 0,1, and A′ = 0′,1′ are the two-component
spinors representing the wave function. According to [17], at each point of the space-time we can associate to the
spinor basis a null tetrad (ℓ,n,m,m) obeying the normalization and orthogonality relations
ℓ n=1, m m= 1, ℓ m=ℓ m=n m=n m=0 (3)
· · − · · · ·
and hence, the Dirac equation can be rewritten as [17]
(D+ǫ ρ)P0+(δ+π α)P1 = iµ Q1′, (4)
− − ∗
(δ+β τ)P0+(∆+µ γ)P1 = iµ Q0′, (5)
− − − ∗
′ ′
(D+ǫ ρ)Q0 +(δ+π α)Q1 = iµ P1, (6)
− − ∗
′ ′
(δ+β τ)Q0 +(∆+µ γ)Q1 = iµ P0, (7)
− − − ∗
where
D =ℓi∂ , ∆=ni∂ , δ =mi∂ , δ =mi∂ , i=1, ,4 (8)
i i i i
···
3
and the spin coefficients are given by
κ = γ , σ =γ , λ=γ , ν =γ , ρ=γ , µ=γ , (9)
(3)(1)(1) (3)(1)(3) (2)(4)(4) (2)(4)(2) (3)(1)(4) (2)(4)(3)
1 1
τ = γ , π =γ , ǫ= γ +γ , γ = γ +γ , (10)
(3)(1)(2) (2)(4)(1) 2 (2)(1)(1) (3)(4)(1) 2 (2)(1)(2) (3)(4)(2)
1 1 (cid:2) (cid:3) (cid:2) (cid:3)
α = γ +γ , β = γ +γ (11)
2 (2)(1)(4) (3)(4)(4) 2 (2)(1)(3) (3)(4)(3)
with (a) denoting t(cid:2)he tetrad index and(cid:3)a=1, ,(cid:2)4. The Ricci rotation(cid:3) coefficients are expressed as
···
1
γ = λ +λ λ , λ = e e e ie j, (12)
(a)(b)(c) 2 (a)(b)(c) (c)(a)(b)− (b)(c)(a) (a)(b)(c) (b)i,j − (b)j,i (a) (c)
where (cid:2) (cid:3) (cid:2) (cid:3)
ℓ1 ℓ2 ℓ3 ℓ4
n1 n2 n3 n4
(e(a)i)= m1 m2 m3 m4  (13)
 m1 m2 m3 m4 
 
 
andits inverseisdenotedby(e(b) ). We recallthattetradindices labelrowsandtensorindices labelcolumns. Taking
i
into account that e =g e k and assuming a diagonal metric tensor g we find that
(a)i ik (a)
ℓ ℓ ℓ ℓ
1 2 3 4
n n n n
(e i)= 1 2 3 4 (14)
(a) m m m m 
1 2 3 4
m m m m
 1 2 3 4 
 
We have different null tetrads leading to different expressions of the spin coefficients. For instance, [17, 18] use the
so-called Kinnersley tetrad
r2 1 ∆ 1 i
ℓ=e = ,1,0,0 , n=e = , r,0,0 , m=e = 0,0, , (15)
(1) (cid:18)∆r (cid:19) (2) (cid:18)2 −2r2 (cid:19) (3) (cid:18) r√2 r√2sinϑ(cid:19)
with∆ =r2 2Mr. Furthermore,onecanalsousetheCartertetrad[19,20]whichallowsforamoreeleganttreatment
r
−
Vierbein κ σ λ ν ρ µ τ π ǫ γ β α
Kinnersley (15) 0 0 0 0 −1r 2M2r−2r 0 0 0 2Mr2 2c√ot2ϑr −β
Carter (17) 0 0 0 0 √∆r ρ 0 0 − M ǫ cotϑ −β
r2√2 2√2r√∆r 2√2r
TABLE I:Table of spin coefficients for different choices of thenull tetrad
oftheseparationproblemandleadstoasimplerformoftheradialsystemthanthosederivedin[17,18]. Tothispurpose
wefirstintroduceavierbein e witha=1, ,4suchthatg =η e(a) e(b) withη =diag(1, 1, 1, 1).
(a) (a)(b) (a)(b)
{ } ··· ⊗ − − −
For the SchwarzschildBH metric we have
√∆ e r e e
e(1) = r,0,0,0 , e(2) = 0, ,0,0 , e(3) =(0,0,r,0), e(4) =(0,0,0, rsinϑ). (16)
r √∆ −
(cid:18) (cid:19) (cid:18) r (cid:19)
With the heelp of (5.119) in [21] we caen construct the followingesymmetric null tetread [11, 22]
r 1 ∆ r 1 ∆ 1 i
ℓ= , r,0,0 , n= , r,0,0 , m= 0,0, , . (17)
√2∆r −rr 2 ! √2∆r rr 2 ! (cid:18) r√2 r√2sinϑ(cid:19)
ThecorrespondingspincoefficientsarelistedinTableI.Wecompletethissectionbygivinganoverviewofthedifferent
butequivalentradialsystemsemergingfromtheDiracequationafterseparationofvariableswhenthe Kinnersleyand
Carter tetrads are used. First of all, [17] separates the Dirac equation in the Schwarzschild BH metric by choosing
the Kinnersley tetrad and by making the ansatz
ei(ωt+kϕ)
P0 = R (r)S (ϑ), P1 =ei(ωt+kϕ)R (r)S (ϑ), (18)
+ +
r√2 − −
′ e i(ωt+kϕ) ′
Q0 = − R (r) S (ϑ), Q1 =e i(ωt+kϕ)R (r) S (ϑ), (19)
+ − +
− r√2 − −
4
where ω is the energy of the particle andk = 1/2, 3/2, is the azimuthalquantum number [11]. This procedure
± ± ···
leads to the following radial and angular systems
(cid:18)−(λ−D0imer) −√(∆λr+D0†im√e∆rr) (cid:19)(cid:18)RR−+ (cid:19)=0, Lλ†12 −Lλ12 !(cid:18)SS−+ (cid:19)=0, (20)
where
d ωr2 d ωr2 d k 1 d k 1
D0 = dr +i∆r , D0† = dr −i∆r , L12 = dϑ + sinϑ + 2cotϑ, L†12 = dϑ − sinϑ + 2cotϑ. (21)
The spectrum of the angular problem is purely discrete and the eigenvalues are given as follows [23]
1
λ (k)=sgn(j) k + j , j Z 0 . (22)
j
| |− 2 | | ∈ \{ }
(cid:18) (cid:19)
Finally, the radial system can be rewritten as
dR ωr
− +i R = (λ+imer)R+, (23)
dr r 2M −
−
dR r M ωr λ im r
+ e
+ − i R = − R . (24)
dr r2 2Mr − r 2M + r2 2Mr −
(cid:18) − − (cid:19) −
The Dirac equation in the Schwarzschild BH metric has been also separated in [18] where the Kinnersley tetrad has
been adopted but the initial ansatz for the spinor is somewhat different from that made in [17]. In this case we let
ei(ωt+kϕ) ei(ωt+kϕ)
P0 = H (r)S (ϑ), P1 = H (r)S (ϑ), (25)
1 1 2 2
r r
′ e i(ωt+kϕ) ′ e i(ωt+kϕ)
Q0 = − H (r) S (ϑ), Q1 = − H (r) S (ϑ). (26)
1 2 2 1
− r r
and after substitution into the Dirac equation we obtain the following radial and angular systems
r√2D0 λ−imer H1 =0, L+ λ S1 =0, (27)
λ+imer r∆√r2 D0†+ ∆Mr !(cid:18)H2 (cid:19) (cid:18)−λ L− (cid:19)(cid:18)S2 (cid:19)
(cid:16) (cid:17)
where D0 and D0† are defined as in [17]. Moreover, L+ = L†21 and L− = L21. Finally, the radial system can be
rewritten as
dH ωr λ
1
+i H = iµ H , (28)
1 2
dr r 2M ∗− r√2
− (cid:18) (cid:19)
dH M ωr 2r λ
2
+ i H = iµ + H . (29)
dr r2 2Mr − r 2M 2 −r 2M ∗ r√2 1
(cid:18) − − (cid:19) − (cid:18) (cid:19)
We endthis sectionwith a shortderivationof the radialsystemwhenthe Cartertetradis chosen. The corresponding
spin coefficients are given in Table I. Let
ei(ωt+kϕ) ei(ωt+kϕ)
P0 = R (r)S (ϑ), P1 = R (r)S (ϑ), (30)
√r√4∆r + + √r√4∆r − −
′ e i(ωt+kϕ) ′ e i(ωt+kϕ)
Q0 = − R (r) S (ϑ), Q1 = − R (r) S (ϑ). (31)
− √r√4∆r − + √r√4∆r + −
Then, the Dirac equation splits into the following radial and angular systems
1
(cid:18)−(√λ∆−riDm0er) −(√λ∆+riDm0†er) (cid:19)(cid:18)RR−+ (cid:19)=0, Lλ2 −L†λ12 !(cid:18)SS−+ (cid:19)=0, (32)
5
where D0, D0†, L21, and L†12 are defined as in [17]. The following property of the radial spinors
R (r)=R (r), R (r)=R (r) (33)
+ +
− −
becomes evident if we rewrite the radial system as
dR ωr λ+im r
e
− +i R = R+, (34)
dr r 2M − √r2 2Mr
− −
dR ωr λ im r
+ e
i R = − R . (35)
+
dr − r 2M √r2 2Mr −
− −
Note that property (33) could not have become transparent if we would have adopted the Kinnersley tetrad. This
also implies that the solution of the Dirac equation will read
ei(ωt+kϕ) ei(ωt+kϕ)
P0 = R (r)S (ϑ), P1 = R (r)S (ϑ), (36)
√r√4∆r + + √r√4∆r − −
′ e i(ωt+kϕ) ′ e i(ωt+kϕ)
Q0 = − R (r) S (ϑ), Q1 = − R (r) S (ϑ). (37)
− √r√4∆r + + √r√4∆r − −
If we let
F F ′ G ′ G
P0 = 1 , P1 = 2 , Q0 = 2 , Q1 = 1 , (38)
√r√4∆ √r√4∆ −√r√4∆ √r√4∆
r r r r
the Dirac equation in the presence of a Schwarzschild manifold can be written as
( + )Ψ=0, Ψ=(F ,F ,G ,G )T (39)
1 2 1 2
R A
with
imer 0 √∆r + 0 0 0 0
− 0 im r 0D √∆ 0 0 −L0−
R= √∆r − 0 e imer − 0rD− , A= 0 −L0+ 0  (40)
 − 0 D− √∆rD+ − 0 −imer  L+ L0− 0 0 
and differential operators
r2 i 1
=∂ ∂ , =∂ ∂ + cotϑ. (41)
r t ϑ ϕ
D± ± ∆r L± ± sinϑ 2
From (39) we immediately obtain the Dirac equation in Hamiltonian form, namely
∆ m √∆
r e r
i∂ Ψ=HΨ, H =i diag( ∂ ,∂ ,∂ , ∂ ) ∆ A+i B (42)
t r2 − r r r − r − r r r2
p
with
0 0 1 0 0 0 0
0 0 0 1 L0− 0 0
A= 1 0 0 0, B =L0+ 0 0 . (43)
0 1 0 0 0 0 −L0−
   −L+ 
   
According to [6] on the region r > 2M and on the hypersurfaces t = const we can introduce the following inner
product
Ψ,Φ = +∞dr r2 πdϑ sinϑ 2πdϕΨ T(t,r,ϑ,ϕ)Φ(t,r,ϑ,ϕ). (44)
∗
h i ∆
Z2M r Z0 Z0
It is not difficult to verify that the Hamiltonian H is symmetric or formally self-adjoint with respect to the inner
product (44). This in turn will imply that ω R. In what follows we say that a time-periodic solution
∈
Ψ(t,r,ϑ,ϕ)=e iωtΨ (r,ϑ,ϕ), ω R, Ψ =0 (45)
− 0 0
∈ 6
6
of the Dirac equation (42) is a bound state if it is normalizable, that is
Ψ,Ψ = Ψ ,Ψ < , (46)
0 0
h i h i ∞
where , is given by (44). If such a solution exists, we say that ω is an eigenvalue of the Hamiltonian H for the
h· ·i
eigenspinor Ψ and ω represents the energy of the bound state. Furthermore, if we introduce the Chandrasekhar
0
ansatz
R (r)S (ϑ)
+ +
R (r)S (ϑ)
Ψ0(r,ϑ,ϕ)=eikϕ R−(r)S−(ϑ)  (47)
+
R−(r)S (ϑ)
 
 − − 
with k = 1/2, 3/2, , we only need to investigate the radial and angular systems (32). The angular system has
± ± ···
beenthoroughlystudiedin[23]whereeigenvaluesandassociatedeigenfunctionshavebeencomputed. Notethatafter
separation of variables of the Dirac equation ω R will be an energy eigenvalue of (42) if there exists some λ R
∈ ∈
and non trivial solutions
R (r) S (ϑ)
R(r)= R−+(r) , S(ϑ)= S−+(ϑ) , r >2M, ϑ∈(0,π) (48)
(cid:18) (cid:19) (cid:18) (cid:19)
satisfying the normalization conditions
+∞dr r2 R(r)2 < , πdϑ sinϑ S(ϑ)2 < . (49)
∆ | | ∞ | | ∞
Z2M r Z0
By means of (47) it is straightforwardto verify that
Ψ,Ψ =2π +∞dr r2 R(r)2 πdϑ sinϑ S(ϑ)2 < , (50)
h i ∆ | | | | ∞
(cid:18)Z2M r (cid:19)(cid:18)Z0 (cid:19)
whenever (49) is satisfied and moreover,the approachfollowed here allows to reduce the eigenvalue equation HΨ =
0
ωΨ andthenormalizationcondition Ψ ,Ψ toapairofboundaryvalueproblemscoupledbytheseparationconstant
0 0 0
h i
λ. From [23] we already know that the second inequality in (49) is satisfied for the eigenfunctions associated to the
eigenvalues of the angular problem. Therefore,in the next two sections we will investigate whether or not there exist
solutions of the radial problem such that the first inequality in (49) is satisfied.
III. NON EXISTENCE OF FERMIONIC BOUND STATES IN THE SCHWARZSCHILD BH METRIC
[1, 2] claimed that bound states solutions for the Dirac equation in the Schwarzschild BH metric exist. The main
idea behind their method is to study approximated solutions of the radial system close to the event horizon and
asymptotically at space-like infinity. In particular, they obtained an asymptotic energy spectrum by requiring that
the Kummer function describing the radial spinor asymptotically at infinity reduces to a polynomial function. The
emergence of these bound states is a result of the oversimplification of the original system of differential equations
governing the radial problem. This becomes particularly clear, if one studies the original radial system without
introducinganysortofapproximation. Forthisreason,wefirstreducethesystemofdifferentialequationssatisfiedby
theradialspinorstoacoupleofgeneralizedHeunequations(see[12–14]),despitetheclaimsin[1,18]thatsucharadial
systemcannotbe reducedto anyknownequationofmathematicalphysics,andthen, wederivea setofnecessaryand
sufficient conditions for the existence of bound state solutions for spin-1/2 particles in the Schwarzschild geometry.
Finally,weprovethatsuchasetofconditionscanneverbesatisfiedifthe spectralparameterω isreal. Ifω isallowed
tobecomplex,polynomialsolutionsexist,butinthiscaseinsteadofboundstates,resonanceswilloccurinagreement
withthe findingsof[3]. Lastbutnotleast,ournovelapproachleadstoanon-existenceresultinline withthefindings
of [5, 7, 9] where the authors reached the same conclusion in the framework of operator theory. To this purpose, let
Ω =2Mω and µ =2Mm . Furthermore, we introduce the rescaled radial variable ρ =r/(2M) with ρ (1,+ ). If
e
∈ ∞
we set
ρ λ+iµρ
f(ρ)= , g(ρ)= , (51)
ρ 1 ρ2 ρ
− −
p
7
then, the radial system represented by (34) and (35) can be rewritten as follows
dR dR
+
− +iΩf(ρ)R (ρ)=g(ρ)R+(ρ), iΩf(ρ)R+(ρ)=g(ρ)R (ρ) (52)
dρ − dρ − −
and it decouples into the following second order linear differential equations
′ ′
R′′(ρ) g (ρ)R′ (ρ)+ Ω2f2(ρ) g(ρ)2+iΩ f′(ρ) f(ρ)g (ρ) R (ρ) = 0, (53)
− − g(ρ) − " −| | − g(ρ)!# −
R′′(ρ) g′(ρ)R′ (ρ)+ Ω2f2(ρ) g(ρ)2 iΩ f′(ρ) f(ρ)g′(ρ) R (ρ) = 0, (54)
+ − g(ρ) + " −| | − − g(ρ)!# +
whereaprimedenotesdifferentiationwithrespecttotheindependentvariableρ. Inordertobringtheabovedifferential
equationsintotheformofageneralizedHeunequationwemakeapartialfractionexpansionofthecoefficientfunctions
and we obtain
′′ ′
R (ρ)+p (ρ)R (ρ)+q (ρ)R (ρ)=0 (55)
± ± ± ± ±
with
1/2 1/2 1 λ
p (ρ)= + , c = i (56)
± ρ ρ 1 − ρ c ± ∓ µ
− − ±
and
λ2 α β γ
q (ρ) = Ω2 µ2+ + ± + ± + ± , α =2Ω2 λ2 µ2 γ , (57)
± − ρ ρ 1 (ρ 1)2 ρ c ± − − − ±
− − − ±
Ω λΩ
β = Ω2 i , γ = . (58)
± ± 2 ± ∓iλ µ
±
By means of the transformation
R (ρ)=e √µ2 Ω2ρR (ρ) (59)
− −
± ±
the differential equation (55) becomes
e
′′ ′
R (ρ)+P (ρ)R (ρ)+Q (ρ)R (ρ)=0 (60)
± ± ± ± ±
with
e e e
σ α β γ
P (ρ)=p (ρ) 2 µ2 Ω2, Q (ρ)= + ± + ± + ± , (61)
± ± − − ± ρ ρ 1 (ρ 1)2 ρ c
p e− − −e ±
where
µ2 Ω2 µ2 Ω2
σ =λ2 − , α =α − , γ =γ + µ2 Ω2. (62)
− p 2 ± ±− p 2 ± ± −
p
In order to eliminate terms like (ρ 1)−2 in Qe we introduce the transfeormation
− ±
R (ρ)=(ρ 1)δ±R (ρ) (63)
± − ±
with δ C and hence, R must satisfy the deifferential equationb
± ∈ ±
′′ ′
b R (ρ)+p (ρ)R (ρ)+q (ρ)R (ρ)=0 (64)
± ± ± ± ±
with
b b b
1/2 2δ +(1/2) 1 σ α β γ
p (ρ)= + ± 2 µ2 Ω2, q (ρ)= ± + ± + ± + ± , (65)
± ρ ρ 1 − ρ c − − ± ρ ρ 1 (ρ 1)2 ρ c
− − ± p b b− −b −b ±
8
where
δ δ δ δ δ
σ =σ ±, α =α 2δ µ2 Ω2+ ± + ± , β =δ2 ± +β , γ =γ ± . (66)
± − 2 ± ±− ± − 2 c 1 ± ±− 2 ± ± ±− c 1
p ±− ±−
b b e b b e
Observe that β = 0 whenever δ = iΩ. Hence, by introducing the rescaled radial variable ρ = r/(2M) and
employing the a±nsatz ± ±
b
R (ρ)=(ρ 1) iΩe √µ2 Ω2ρR (ρ), (67)
± − −
± − ±
the radial system (34) and (35) decoupled into the following generalizebd Heun equations
′′ 2 1 µn, ′ β0, +β1, ρ+β2, ρ2
R (ρ)+ − ± +α R (ρ)+ ± ± ± R (ρ)=0 (68)
± nX=0 ρ−ρn ! ± 2n=0(ρ−ρn) ±
b b b
Q
where α= 2 µ2 Ω2 and 0,µ , 0,µ , and 0,µ . Here, µ =1/2, µ = (1/2) 2iΩ, µ =2 are
0, 1, 2, 0, 1, 2,
the exponen−ts assoc−iated to th{e simp±l}e s{ingula±ri}ties ρ {=0, ρ±}=1, and ρ±=c with c± = iλ/µ∓while the±point at
0 1 2
infinity is an irpregular singular point of rank at most one. Furthermore, we hav±e ± ∓
β =c σ , β = c (α +σ ) γ σ , β =α +γ +σ (69)
0, 1, 2,
± ± ± ± − ± ± ± − ±− ± ± ± ± ±
with
b b b b b b b b
µ2 Ω2 Ω µ2 Ω2 Ω
α =2Ω2 λ2 µ2 − (1 4iΩ) i , γ = iΩ+ µ2 Ω2, σ =λ2 − i . (70)
± − − − p 2 ± ∓ 2 ± ± − ± − p 2 ∓ 2
p
Loobking back at (67), we see that the problem of existbence of bound states redubces to the question whether or not
the generalizedHeun equation (68) admits polynomial solutions for Ω <µ, since then (67) would clearly satisfy the
| |
normalization condition (49). To this purpose, let us first rewrite (68) as follows
′′ ′
ρ(ρ 1)(ρ c )R (ρ)+P (ρ)R (ρ)+Q (ρ)R (ρ)=0 (71)
− − ± ± ± ± ± ±
with c as defined in the statement of the prevbious theorem abnd b
±
P (ρ) = (1 µ )(ρ 1)(ρ c )+(1 µ )ρ(ρ c )+(1 µ )ρ(ρ 1)+αρ(ρ 1)(ρ c ), (72)
0, 1, 2,
± − ± − − ± − ± − ± − ± − − − ±
Q (ρ) = β +β ρ+β ρ2. (73)
0, 1, 2,
± ± ± ±
Furthermore, suppose that
∞
R (ρ)= d (ρ 1)n. (74)
,n
± ± −
n=0
X
b
In orderthat this infinite seriesstops at some fixed value of n, we first needto derive the recurrencerelationsatisfied
by the coefficients d . To this purpose it results convenient to introduce the independent variable transformation
,n
τ =ρ 1. Then, our±differential equation becomes
−
′′ ′
A(τ)R (τ)+P (τ)R (τ)+Q (τ)R (τ)=0 (75)
± ± ± ± ±
with b b b
A(τ)=τ3+a τ2+a τ, P (τ)=ατ3+b τ2+b τ +b , Q (τ)=β τ2+c τ +c , (76)
1, 2, 1, 2, 3, 2, 1, 2,
± ± ± ± ± ± ± ± ± ±
where
a = 2 c , a =1 c , (77)
1, 2,
± − ± ± − ±
1 1
b = α(2 c ) 2iΩ, b =α(1 c ) c (1 2iΩ)+ 4iΩ, b =(1 c ) 2iΩ , (78)
1, 2, 3,
± − ± ± ± − ± − ± ± 2 ± ± − ± 2 ±
(cid:18) (cid:19)
β = 2Ω2 µ2 2iΩ µ2 Ω2, c =β +2β , c =β +β +β . (79)
2, 1, 1, 2, 2, 0, 1, 2,
± − ∓ − ± ± ± ± ± ± ±
p
9
Substituting R (τ) = ∞n=0d ,nτn into (75) and shifting indices appropriately yield the following four term recur-
rence relation ± ±
P
b
ϕ (n+1)d +ϕ (n)d +ϕ (n 1)d +ϕ (n 2)d =0, n=0,1,2, (80)
1 ,n+1 2 ,n 3 ,n 1 4 ,n 2
± ± − ± − − ± − ···
with
ϕ (ξ) = a ξ(ξ 1)+b ξ, ϕ (ξ)=a ξ(ξ 1)+b ξ+c , (81)
1 2, 3, 2 1, 2, 2,
± − ± ± − ± ±
ϕ (ξ) = ξ(ξ 1)+b ξ+c , ϕ (ξ)=αξ+β . (82)
3 1, 1, 4 2,
− ± ± ±
In order to find under which conditions we may have polynomial solutions of the form
N
d τn, N =0,1,2, (83)
,n
± ···
n=0
X
let n=N +2 in (80). Then, we obtain
ϕ (N +3)d +ϕ (N +2)d +ϕ (N +1)d +ϕ (N)d =0, N =0,1,2, (84)
1 ,N+3 2 ,N+2 3 ,N+1 4 ,N
± ± ± ± ···
For any fixed N we will have a polynomial solution of degree N whenever
ϕ (N)=0, d =d =0. (85)
4 ,N+1 ,N+2
± ±
Bound states solutions will exist for those realvalues of Ω suchthat Ω <µ and Ω satisfies simultaneouslythe above
| |
set of equations. It is not difficult to verify that the condition ϕ (N) = 0 can be rewritten as αN +β = 0 and
4 2,
±
taking into account that α= 2 µ2 Ω2 the same condition reads
− −
p
2Ω2 µ2 2N µ2 Ω2 = 2iΩ µ2 Ω2. (86)
− − − ± −
We argue now by contradiction. Let us suppose thpat for some N Np 0 there exists a bound state with energy
Ω R such that Ω < µ. This would imply that the l.h.s. of∈(86)∪r{ep}resenting a real number should be equal
N N
∈ | |
to the r.h.s. of (86) representing an imaginary number and hence, we have a contradiction. Therefore, there are no
polynomial solutions for any N =0,1,2, .
···
IV. A DEFICIENCY INDEX APPROACH
In this section we offer an alternative proof that no bound states exist for the Dirac equation in the Schwarzschild
BH metric. This is achieved by constructing a suitable transformation of the radial system and showing that the
deficiency indices of the transformed radial operator are zero. This is not surprising since the deficiency index of a
differentialoperatorcountsthenumberofsquareintegrablesolutions. InordertobeabletoapplytheIndexTheorem
in [8] we bring the radial system (32) into the form of a symmetric Dirac system. This is done by transforming the
radial spinors according to
(R (r),R (r))T =(F(r) iG(r),F(r)+iG(r))T (87)
+
− −
where T denotes transposition and introducing a tortoise coordinate defined through
du r
= (88)
dr r 2M
−
whose solution is
u(r)=r+2Mln(r 2M). (89)
−
Note that u + as r + and u as r 2M+. If we set Φ=(F,G)T, the system (32) becomes
→ ∞ → ∞ →−∞ →
dΦ b
( Φ)(u):=J +B(u)Φ=ωΦ (90)
U du
b
b b b
10
with
J = 0 1 , B(u)= ∆r(u) mer(u) λ . (91)
(cid:18)−1 0 (cid:19) pr2(u) (cid:18) λ −mer(u)(cid:19)
According to the above transformations the integrability condition (49) for the radial spinors simplifies to
+
(Φ,Φ)= ∞du(F2(u)+G2(u))<+ . (92)
∞
Z−∞
The formal differential operator is fobrmbally symmetric because J = J and B = B . Let S be the minimal
∗ ∗ min
operator associated to such thaUt S acts on the Hilbert space L2(R−,du) with respect to the scalar product (, )
min
U · ·
introduced in (92). Then, the operatorS with domainof definition D(S )=C (R)2 such that S Φ:= Φ
min min 0∞ min U
for Φ D(S ) is densely defined and closable. Let S denote the closure of S . In order to apply Neumark’s
min min
∈
decompositionmethod[24]weintroduceminimaloperatorsS associatedto whenthelatterisrestrictebdtothbe
min,
half-blines I+ = [0,+ ) and I = ( ,0], respectively. Moreov±er, we considerUSmin, acting on the Hilbert space
L2(I ,du) with resp∞ect to the−scala−r∞product (, ). The operators S with domai±ns of definition D(S ) =
min, min,
± · · ± ±
C (I )2 andsuchthatS Φ := Φ forΦ D(S )areagaindenselydefinedandclosable. Furthermore,
0∞is in± the limit point casmein(,l±.p.c±.) atU ±. Thi±s ∈can bemseine,n±as follows. First of all, we recall that since the limit
U ±∞
point and limit circle cases are mbutuallybexclusibve, we can determine the appropriate case if we examine the solution
of (90) for a single value of ω. Hence, without loss of generality we set ω =0 and consider the system
dΨ
J +B(u)Ψ=0. (93)
du
b
For u it is straightforwardto verify that the radial varbiable r admits the asymptotic expansion
→−∞
u u
r=2M +e2M + eM . (94)
O
Furthermore, in the same asymptotic limit the matrix B(u) has t(cid:0)he fo(cid:1)llowing decomposition
B(u)=B0(u)+B1(u), B0(u)=e4uM (cid:18) m2λMe −2mλMe (cid:19), |B1(u)|≤Ce43Mu , (95)
where C is a positive constant. Finally, it can be easily checked that the system (93) has asymptotic solution for
u given by
→−∞
u 1 3u u 0 3u
F(u)=exp 2γe2M + e4M , G(u)=exp 2γe2M + e4M , γ = λ2+4M2m2. (96)
0 O − 1 O e
(cid:0) (cid:1)(cid:18) (cid:19) (cid:16) (cid:17) (cid:0) (cid:1)(cid:18) (cid:19) (cid:16) (cid:17) p
When u + , the tortoise coordinate becomes
→ ∞
1
u=r+2Mlnr+ (97)
O u
(cid:18) (cid:19)
from which we find that r can be expressed as a function of u as follows
u
e2M u 1
r(u)=exp W + . (98)
− 2M − 2M O u
(cid:18) (cid:20) (cid:18) (cid:19) (cid:21)(cid:19) (cid:18) (cid:19)
Here, W denotes the so-called Lambert function [25]. Finally, expanding W asymptotically for u + as in [26]
→ ∞
yields
1
r =u 2Mlnu+ . (99)
− O u
(cid:18) (cid:19)
When u + the matrix B(u) admits the decomposition
→ ∞
m 0 D
B(u)=B (u)+B (u), B (u)= e , B (u) (100)
0 1 0 0 me | 1 |≤ u
(cid:18) − (cid:19)
e e e e